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In this work the authors deal with linear second order partial
differential operators of the following type \[
H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x)
X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)\] where
\(X_{1},X_{2},\ldots,X_{q}\) is a system of real
Hörmander's vector fields in some bounded domain
\(\Omega\subseteq\mathbb{R}^{n}\), \(A=\left\{ a_{ij}\left(
t,x\right) \right\} _{i,j=1}^{q}\) is a real symmetric uniformly
positive definite matrix such that
\[\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x)
\xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q},
x \in\Omega,t\in(T_{1},T_{2})\] for a
suitable constant \(\lambda>0\) a for some real numbers
\(T_{1} < T_{2}\).

In this work the authors deal with linear second order partial
differential operators of the following type \[
H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x)
X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)\] where
\(X_{1},X_{2},\ldots,X_{q}\) is a system of real
Hörmander's vector fields in some bounded domain
\(\Omega\subseteq\mathbb{R}^{n}\), \(A=\left\{ a_{ij}\left(
t,x\right) \right\} _{i,j=1}^{q}\) is a real symmetric uniformly
positive definite matrix such that
\[\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x)
\xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q},
x \in\Omega,t\in(T_{1},T_{2})\] for a
suitable constant \(\lambda>0\) a for some real numbers
\(T_{1} < T_{2}\).